Undergraduate Courses 2017-18

Differentiation in several variables, with applications in approximation, maximum and minimum and geometry. Integration in several variables, vector analysis.

MATH1001

MATH1002

This course teaches basic application techniques in single-variable calculus and linear algebra. Key topics include: systems of linear equations and matrices, functions and graphing, derivatives and optimization, integration and applications.

Sequences, series, gradients, chain rule. Extrema, Lagrange multipliers, line integrals, multiple integrals. Green's theorem, Stroke's theorem, divergence theorem, change of variables.

This is a one-year course with focus on limits, one variable calculus, sequences, series, gradients, chain rule, extrema, Lagrange multipliers, line integrals, multiple integrals, Jacobians, Implicit function theorem, Green's theorem, Stoke's theorem, divergence theorem.

This course teaches basic application techniques in multivariable calculus and concepts in probability. Key topics include: vectors and vector valued functions, functions of several variables, partial derivatives, constrained optimization, multiple integrals, and basic probability.

Expository lectures and discussion on basic mathematical concepts and ideas, historical developments in various areas of mathematics, and selected trends and advances in mathematical sciences. Third year and fourth year students require instructor's approval to take the course.

Vector space, matrices and system of linear equations, linear mappings and matrix forms, inner product, orthogonality, eigenvalues and eigenvectors, symmetric matrix.

Systems of linear equations; vector spaces; linear transformations; matrix representation of linear transformations; linear operators, eigenvalues and eigenvectors; similarity invariants and canonical forms.

A first course in calculus. Sets and logic, functions, limit and continuity, differentiation, applications and problem solving.

Logic: propositions, axiomatization of propositional calculus, deduction theorem, completeness and soundness. Combinatorics: permutations and combinations, generating functions. Set theory: basic operations on sets, relations, countable and uncountable sets. Third year and fourth year students require instructor's approval to take the course.

A second course in calculus. Integration, applications of definite integral, techniques of integration, sequences and series, parametric equations.

A systematic introduction to statistical inference, including the necessary probabilistic background, point and interval estimation, hypothesis testing.

First order equations, second order equations, Laplace transform method, numerical solution of initial value problems, boundary-value problems.

First and second order differential equations, initial value problems, series solutions, Laplace transform, numerical methods, boundary value problems, eigenvalues and eigenfunctions, Sturm-Liouville theory.

First order equation, linear second order equations, Laplace transform, Euler and Runge-Kutta methods, introduction to partial differential equations, matrix, systems of linear equations, eigenvalue and eigenvector, systems of differential equations, orthogonal projection.

The purpose of this course is to expose students to the role of mathematics in the development and maintenance of civilization. This course examines how mathematics shaped and was shaped by the course of human events. The course will cover the growth, development and far-reaching applications of trigonometry, navigation, cartography, logarithms, algebra, and calculus through ancient, medieval, post-Renaissance and modern times. Topics include: Number systems and the invention of positional notation, Egyptian arithmetic, Babylonian algebra, Greek trigonometry, the contribution of Chinese mathematics etc.

Discussions on problem solving techniques. Basics materials in combinatorics, number theory, geometry and mathematical games.

Sets and functions, real numbers, limits of sequences and series, limits of functions, continuous functions, differentiation, Riemann integration, additional topics.

This is a one-year course with focus on sets and functions, real numbers, open and closed sets, limits of sequences and series, limits and continuity of functions, Taylor's series, differentiations, Riemann integrations, uniform convergence.

The MATH 203 - 204 series is a rigorous sequence in analysis on the line and higher dimensional Euclidean spaces. Limit, continuity, least upper bound axiom, open and closed sets, compactness, connectedness, differentiation, uniform convergence, and generalization to higher dimensions. Enrollment in the course requires approval of the course instructor.

The MATH 203 - 204 series is a rigorous sequence in analysis on the line and higher dimensional Euclidean spaces. Differentiation and integration in higher dimensions, implicit function and inverse function theorem, Stokes theorem, and Lebesgue measure.

This course teaches fundamental concepts in calculus and provides mathematical preparation for students who are going to take further courses in mathematics. Key topics include: logic and sets, functions, limits and continuity, differentiation and graphing, integration; improper integrals, sequence and series, power and Taylor series.

The MATH 217 - 218 sequence is a rigorous introduction to linear algebra and abstract algebra. Vector spaces over the fields of real numbers and complex numbers, linear transformations, geometry, groups, bases, abstract fields, rings, change of bases, spectral theorems.

The MATH 217 - 218 sequence is a highly rigorous introduction to linear algebra and abstract algebra. Groups, rings, homomorphisms, quotients, group actions, polynomial rings, Chinese remainder theorem, field extensions.

This is a course designed for students entering through various special schemes. Key topics include: algebra, functions, limit, derivatives, optimization problems, elementary transcendental functions, integration and applications.

The MATH 023 - 024 sequence is designed for preparatory-year students who are going to take further courses in mathematics. Topics include equations on lines and planes, parametric equations, polar coordinates, functions, limits and continuity. Differential calculus, curve sketching, concavity, applications which includes optimization problems, physical applications, etc. Some rigorous theoretical results on limits, continuity and differentiation will be discussed.

Computer arithmetric, matrix computation, interpolation and approximation, numerical integration, solution of nonlinear equations.

Basic numerical analysis, including stability of computation, linear systems, eigenvalues and eigenvectors, nonlinear equations, interpolation and approximation, numerical integration and solution of ordinary differential equations, optimization. Fortran may also be taught.

An introduction to combinatorics: What is combinatorics? Permutations and combinations, binomial theorem, generating permutations and combinations, pigeonhole principle, Ramsey theory, inclusion-exclusion principle, rook polynomials, linear recurrence relations, nonhomogeneous linear recurrence relations of the first and second order, generating functions, Catalan numbers, Striling numbers, partition numbers, matchings and stable matchings, systems of distinctive representatives, block designs, Steiner triple systems, Latin squares, Burnside's lemma, Polya counting formula.

The MATH 023 - 024 sequence is designed for preparatory-year students who are going to take further courses in mathematics. Topics include integral calculus, techniques of integration, improper integrals, applications of integrals, infinite series. Some rigorous theoretical results on integration and infinite series will be discussed.

Sample spaces, conditional probability, random variables, independence, discrete and continuous distributions, expectation, correlation, moment generating function, distributions of function of random variables, law of large numbers and limit theorems.

Sampling theory, order statistics, limiting distributions, point estimation, confidence intervals, hypothesis testing, non-parametric methods.

Focuses on a coherent collection of topics selected from a particular branch of mathematics. A student may repeat the course for credit if the topics studied are different each time.

Functions of several variables, implicit and inverse function theorem, uniform convergence measure and integral on the real line.

Existence and uniqueness theorems of ordinary differential equations, theory of linear systems, stability theory, study of singularities, boundary value problems.

Complex differentiability; Cauchy-Riemann equations; contour integrals, Cauchy theory and consequences; power series representation; isolated singularities and Laurent series; residue theorem; conformal mappings.

Introduction to manifolds, metric spaces, multi-linear Algebra, differential forms, Stokes theorem on manifolds, cohomology.

Derivatives of the Laplace equations, the wave equations and diffusion equation; Methods to solve equations: separation of variables, Fourier series and integrals and characteristics; maximum principles, Green's functions.

Lagrangian and Eulerian methods for the flow description; derivation of the Euler and Navier-Stokes equations; sound wave and Mach number; 2D irrotational flow; elements of aerofoil theory; water wave dispersion relation; shallow water waves; ship wave pattern; dynamics of real fluid, stokes flow and boundary layer theory.

Working in small teams, students are required to select a topic in pure mathematics, applied mathematics or statistics area. They will discuss and write up their learning and present it at the seminars. The level of the topics can range from simple calculus to advanced topology, geometry or statistics. Students may repeat the course for credit at most two times.

Zero-sum games; minimax theorem; games in extensive form; strategic equilibrium; bi-matrix games; repeated Prisonner's Dilemma; evolutionary stable strategies; games in coalition form; core; Shapley Value; Power Index; two-side matching games.

Polynomials; Jordan canonical form, minimal polynomials, rational canonical form; equivalence relation; group, coset, group action; introduction to rings and fields.

Groups and symmetry. Group actions. Symmetries of pictures, graphs, Euclidean spaces, platonic solids. Polynomials, field extensions, impossibility of certain geometric constructions. Finite fields. Applications to cryptography.

Prime numbers, unique factorization, modular arithmetic, quadratic number fields, finite fields, p-adic numbers, coding theory, computational complexity.

General linear groups, orthogonal groups, unitary groups, symplectic groups, exponential maps, maximal tori, Clifford algebra, spin groups.

Axioms and models, Euclidean geometry, Hilbert axioms, neutral (absolute) geometry, hypernbolic geometry, Poincare model, independence of parallel postulate.

Curve theory; curvature and torsion, Frenet-Serret frame; surface theory: Weingarten map, first and second fundamental forms, curvatures, Gaussian map, ruled surface, minimal surface; instrinsic geometry: Theorema Egregium, Coddazi-Mainardi equations, parallel transport, geodesics, exponential map, Gauss-Bonnet theorem.

Metric, topology, continuous map, Hausdorff, connected, compact, graph, Euler number, CW-complex, classification of surfaces.

Introduction to finite difference and finite element methods for the solution of elliptic, parabolic and hyperbolic partial differential equations; including the use of computer software for the solution of differential equations.

Introduction to fundamental principles and spirit of applied mathematics. Emphasis on manner in which mathematical models are constructed for physical problems. Illustrations from many fields of endeaver, e.g. physical science, biology, economics, traffice dynamics.

Discrete time Markov chains and the Poisson processes. Additional topics include birth and death process, elementary renewal process and continuous-time Markov chains.

Estimation and hypothesis testing in linear regression, residual analysis, multicollinearity, indicator variables, variable selection, nonlinear regression.

Computer-oriented statistical analysis including generalized linear models, classification, principal component analysis, survival analysis, binary data. Real data sets presented for analysis using statistical packages such as SAS, Minitab and S-plus.

The sign test; Wilcoxon signed rank test; Wilcoxon rank-sum test; Kruskal-Wallis test; rank correlation; order statistics; robust estimates; Kolmogorov-Smirnov test; nonparametric curve estimation.

Basic and standard sampling design and estimation methods. Particular attention given to variance estimation in sampling procedures. Topics include: simple random sampling, unequal probability sampling, stratified sampling, ratio and subpopulation and multistage designs.

Inferences of means and covariance matrices, canonical correlation, discriminant analysis, multivariate ANOVA, principal components analysis, factor analysis.

Stationarity, (partial) auto-correlation function, ARIMA modeling, order selection, diagnostic, forecasting, spectral analysis.

Random walk models for asset price and interest rate processes. Risk neutral valuation principle, binomial model. Lattice tree algorithms for pricing options. Monte Carlo simulation techniques. Yield curve fitting, no-arbitrage interest rate models. Pricing algorithms for embedded features in fixed income instruments.

Discrete securities models. Concept of arbitrage. Risk neutral probability measures, valuation of continge claims, complete and incomplete markets. Optimal consumption and investment problems. Actuarial stochastic investment models, insurance and pension applications. Risk theory, value at risk, ruin probability.

Population, ecology, infectious disease, genetic, and biochemistry models. Additional topics chosen by instructor.

Examples and properties of metric spaces. Contractive mapping theorem, Baire category theorem, Stone-Weierstrass theorem, Arzela-Ascoli theorem. Properties of normed spaces and Hibert spaces. Riesz theorem. Completeness of Lp functions, continuous functions and functions of bounded variations. Best approximation theorem on Hilbert space.

Topological vector spaces. Hahn-Banach theorem, open mapping theorem, closed graph theorem, uniform boundedness theorem, separation theorem, Krein-Milman theorem. Weak topologies and reflexivity. Adjoints and duality. Compact and Fredholm operators with index. Normal operators. Spectral theorem for compact normal operators.

Supplementary study of specialized topics for students of pure mathematics.

Supplementary study of specialized topics for students of applied mathematics.

Supplementary study of specialized topics for students of statistics.

Undergraduate research conducted under the supervision of a faculty member. A written report and presentation are required. Scope may include (i) identifying a non-reference problem and proposing methods of solutions, and (ii) acquiring a specific research skill. Students may take MATH 397 or/and MATH 398 for credits up to two times.

Work in any area of mathematics under the guidance of a faculty member. The project either surveys a research topics or describes a small project completed by the student.

Independent work in an area of mathematics under the guidance of a faculty member. The project either surveys a research topic or describes a small project completed by the student. For MATH and MAEC Year 4 students only.

The course prepares the students for taking a course in calculus. The focus is on basic algebra and analytic geometry: equations and inequalities, systems of linear equations, factoring, conic sections, logarithmic and exponential functions, elementary finite mathematics.

This is a general science course that introduces students to selected disciplines or topics of high popular interest. The crucial roles that mathematics play are emphasized. Materials are chosen to enrich and enhance students' appreciation of science and mathematics.

For students in the Science School only. A practical training course for a total duration of two weeks covering basic PC hardware architecture, an introduction to Windows 2000/XP operating systems and web based learning application software.